# Compute the Fourier expansion of adelic Eisenstein series associated to the classical holomorphic Eisenstein series.

• For each place $$v$$ of $$\mathbf{Q}$$, define $$\Phi_v:(\mathbf{Q}_v)^2\to\mathbf{C}$$ by $$\Phi_v(x,y)=\begin{cases} \mathbb{I}_{\mathbf{Z}_v}(x)\mathbb{I}_{\mathbf{Z}_v}(y)&\text{if v<\infty},\\ 2^{-k}(x+\sqrt{-1}y)^ke^{-\pi(x^2+y^2)}&\text{if v=\infty}. \end{cases}$$ where $$\mathbb{I}_{\mathbf{Z}_v}$$ is the characteristic function of $$\mathbf{Z}_v$$.

• Define $$f_{v,s}:\mathrm{GL}_2(\mathbf{Q}_v)\to\mathbf{C}$$ for $$s>>0$$ by $$f_{v,s}(g_v)=\lvert{\det(g_v)}\rvert^{s+\frac{1}{2}}\int_{\mathbf{Q}_v^\times}\Phi_v((0,t)g_v)\lvert t_v\rvert^{2s+1}d^\times t_v$$ where the Haar measure are normalized so that $$\int_{\mathbf{Q}_v^\times}\mathbb{I}_{\mathbf{Z}_p^\times}(t_v)d^\times t_v=1$$. Put $$f_s=\bigotimes_vf_{v,s}$$, which has analytic continuation to all $$s\in\mathbf{C}$$.

• Define the Eisenstein series $$\varphi_k:\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})\to\mathbf{C}$$ by $$\varphi_k(g)=\sum_{\gamma\in B\backslash G}f_s(\gamma g)|_{s=(1-k)/2}$$ for positive even integers $$k\geq 4$$, where $$G=\mathrm{GL}_2(\mathbf{Q})$$ and $$B$$ is the subgroup of upper triangular matrices in $$G$$. I believe that $$y^{-k/2}\varphi_k\begin{pmatrix}y&x\\0&1\end{pmatrix}=E_k(z):=-\frac{B_{k}}{2 k}+\sum_{n=1}^{\infty} \sigma_{k-1}(n) q^{n},\quad z=x+\sqrt{-1}y$$ which is the classical Eisenstein series of weight $$k$$. This can be verified by the Fourier expansion $$\varphi_k(g)=\left.f_s(g)+\sum_{\alpha\in\mathbf{Q}}\prod_v\int_{\mathbf{Q}_v}f_{v,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&w\\0&1\end{pmatrix}g_v\right)\psi(-\alpha w)dw\right|_{s=(1-k)/2}.$$ However, these are the results of my computations:

• $$f_s\begin{pmatrix}y&x\\0&1\end{pmatrix}|_{s=(1-k)/2}=0$$.

• $$\int_{\mathbf{Q}_v}f_{\infty,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\begin{pmatrix}y&x\\0&1\end{pmatrix}\right)\psi(-\alpha w)dw|_{s=(1-k)/2}=0$$ if $$\alpha<0$$.

• $$\int_{\mathbf{Q}_v}f_{\infty,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\begin{pmatrix}y&x\\0&1\end{pmatrix}\right)\psi(-\alpha w)dw|_{s=(1-k)/2}=y^{k/2}e^{2\pi\sqrt{-1}\alpha z}$$ if $$\alpha\geq 0$$.

For $$v<\infty$$:

• $$\int_{\mathbf{Q}_v}f_{v,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\right)dw|_{s=(1-k)/2}=\zeta(1-k)$$.

• $$\int_{\mathbf{Q}_v}f_{v,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\right)\psi(-\alpha w)dw|_{s=(1-k)/2}=\sum_{l=0}^{\mathrm{ord}_v\alpha}v^{l(k-1)}$$ if $$\alpha\neq 0$$.

Hence $$y^{-k/2}\varphi_k\begin{pmatrix}y&x\\0&1\end{pmatrix}=\zeta(1-k)+\sum_{\alpha\in\mathbf{Q}^\times, \alpha>0} e^{2\pi\sqrt{-1}\alpha z}\prod_{v<\infty}\sum_{l=0}^{\mathrm{ord}_v\alpha}v^{l(k-1)}=\color{red}{-\frac{B_k}{k}}+\sum_{n=1}^{\infty}\sigma_{k-1}(n) q^{n},$$ which is not what I expect. Where is the factor of 2 missing in my calculation? (Or my calculation is correct and $$y^{-k/2}\varphi_k\begin{pmatrix}y&x\\0&1\end{pmatrix}\neq E_k(z)$$ in fact.)